ODE
\[ -a y(x) y'(x)+b+x y'(x)^2=0 \] ODE Classification
[[_homogeneous, `class G`], _rational, _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.635193 (sec), leaf count = 243
\[\left \{\text {Solve}\left [\frac {-2 a \tanh ^{-1}\left (\frac {\sqrt {a^2 y(x)^2-4 b x}}{a y(x)}\right )-2 (a-1) \tanh ^{-1}\left (\frac {\sqrt {a^2 y(x)^2-4 b x}}{y(x)-a y(x)}\right )+a \log \left ((1-2 a) y(x)^2+4 b x\right )-\log \left ((1-2 a) y(x)^2+4 b x\right )+a \log (4 b x)}{2 a-1}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a^2 y(x)^2-4 b x}}{a y(x)}\right )+2 (a-1) \tanh ^{-1}\left (\frac {\sqrt {a^2 y(x)^2-4 b x}}{y(x)-a y(x)}\right )+a \log \left ((1-2 a) y(x)^2+4 b x\right )-\log \left ((1-2 a) y(x)^2+4 b x\right )+a \log (4 b x)}{2 a-1}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.023 (sec), leaf count = 106
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\it \_T}}^{{\frac {1}{a} \left ( 1-{a}^{-1} \right ) ^{-1}}} \left ( {\frac {b}{{{\it \_T}}^{ \left ( a-1 \right ) ^{-1}} \left ( 2\,a-1 \right ) {{\it \_T}}^{2}}}+{\it \_C1} \right ) ,y \left ( {\it \_T} \right ) ={\frac {1}{{\it \_T}\,a{{\it \_T}}^{ \left ( a-1 \right ) ^{-1}} \left ( 2\,a-1 \right ) } \left ( 2\, \left ( {\it \_C1}\,{{\it \_T}}^{ \left ( a-1 \right ) ^{-1}}{{\it \_T}}^{2}+b \right ) \left ( a-1/2 \right ) {{\it \_T}}^{ \left ( a-1 \right ) ^{-1}}+{{\it \_T}}^{ \left ( a-1 \right ) ^{-1}}b \right ) }] \right \} \] Mathematica raw input
DSolve[b - a*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[(-2*a*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] - 2*(-1 + a)*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*y[x]^2] + a*Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]], Solve[(2*a*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(a*y[x])] + 2*(-1 + a)*ArcTanh[Sqrt[-4*b*x + a^2*y[x]^2]/(y[x] - a*y[x])] + a*Log[4*b*x] - Log[4*b*x + (1 - 2*a)*y[x]^2] + a*Log[4*b*x + (1 - 2*a)*y[x]^2])/(-1 + 2*a) == C[1], y[x]]}
Maple raw input
dsolve(x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0, y(x),'implicit')
Maple raw output
[x(_T) = _T^(1/a/(1-1/a))*(b/(_T^(1/(a-1)))/(2*a-1)/_T^2+_C1), y(_T) = (2*(_C1*_T^(1/(a-1))*_T^2+b)*(a-1/2)*_T^(1/(a-1))+_T^(1/(a-1))*b)/_T/a/(_T^(1/(a-1)))/(2*a-1)]